Back to . . .  Deposit #53 Curves of Constant Width and Reuleaux Polygons

 Observe the path of the triangle as it rotates.  First, the path of the boundary is not a square at the four corners.  But because the triangle rotates within a square, it is the basis for a square drill bit. The behavior of the center is also fascinating.  The center does not remain fixed and thus traces a path composed of four arcs of an ellipse. This section features the  Constant Width Curves or Orbiform Curves. They are also known as Reuleaux Polygons, most often the triangle,  or "Rollers." Their well-known application is found in the  Wankel Engine

 MATHEMATICA®Code

Historical Sketch:

A constant width curve is a planar convex oval with the property that the distance between two parallel tangents to the curve is constant.  Visualize a circle inscribed in square with the circle rolling, or rotating in the square.  The diameter of the circle is the same as the width of the square.  The width of a closed convex curve is defined to be the distance between the parallel lines bounding it.  The parallel lines of the square are sometimes called "supporting lines."  Please note, the inscribed asteroid does not fit the definition.

Some background is helpful.  Unlike many plane curves, the constant width and Reuleaux polygon investigations are rooted in machine design and engineering.  Moreover, compared to the history of most plane curves, this work is relatively young.

Franz Reuleaux (1829 - 1905) recognized that simple plane curves of constant width might be constructed from regular polygons with an odd number of sides.  Thus, triangles and pentagons are frequently constructed using a corresponding number of intersecting arcs.

In engineering, Felix Heinrch Wankel (1902-1988) designed a rotor engine which has the shape of a Reuleaux triangle inscribed in a chamber, rather than the usual piston, cylinder, and mechanical valves.  The rotor engine, now found in Mazda automobiles has 40% fewer parts and thus far less weight.  Within the Wankel rotor, three chambers are formed by the sides of the rotor and the wall of the housing. The shape, size, and position of the chambers are constantly altered by the rotation of the rotor, i.e., the Reuleaux triangle or deltoid. _

Areas

 Area of the Triangle Using well-known 300-600-900, Area of a Segment

Total Area of a Reuleaux Triangle